Machine Learning Assisted Experimental Characterization of Bubble Dynamics in Gas–Solid Fluidized Beds

This study introduces a machine learning (ML)-assisted image segmentation method for automatic bubble identification in gas–solid quasi-2D fluidized beds, offering enhanced accuracy in bubble recognition. Binary images are segmented by the ML method, and an in-house Lagrangian tracking technique is developed to track bubble evolution. The ML-assisted segmentation method requires few training data, achieves an accuracy of 98.75%, and allows for filtering out common sources of uncertainty in hydrodynamics, such as varying illumination conditions and out-of-focus regions, thus providing an efficient tool to study bubbling in a standard, consistent, and repeatable manner. In this work, the ML-assisted methodology is tested in a particularly challenging case: structured oscillating fluidized beds, where the spatial and time evolution of the bubble position, velocity, and shape are characteristics of the nucleation-propagation-rupture cycle. The new method is validated across various operational conditions and particle sizes, demonstrating versatility and effectiveness. It shows the ability to capture challenging bubbling dynamics and subtle changes in velocity and size distributions observed in beds of varying particle size. New characteristic features of oscillating beds are identified, including the effect of frequency and particle size on the bubble morphology, aspect, and shape factors and their relationship with the stability of the flow, quantified through the rate of coalescence and splitting events. This type of combination of classic analysis with the application of the ML assisted techniques provides a powerful tool to improve standardization and address the reproducibility of hydrodynamic studies, with the potential to be extended from gas–solid fluidization to other multiphase flow systems.

In the bubble tracking model, k 1 and k 2 are critical parameters.For the validation of the model's robustness and the enhancement of its accuracy, an array of k 1 and k 2 values was examined across various cases.As displayed in Figure S1, the statistical results reveal that the case k 1 = 0.64 and k 2 = 1/k 1 = 1.5625 yields a high rate of correct identifications for bubble coalescence and splitting events, with a notably low count of incorrect and undermined bubble events.Consequently, these values were chosen for the bubble tracking model to effectively identify and analyze multi-bubble behaviors.

Bubble tracking algorithm
Figure S2 displays the network of bubble tracking, with three rows representing a sequence of bubble track IDs at three successive time frames: n -1, n, n + 1.This network provides a method for identifying and tracking bubble behaviors between successive time frames.
Regarding binary breakup followed by re-coalescence, a different set of rules for bubble tracking is applied.The arrows among bubble IDs in consecutive frames form a network, illustrating a bubble graph or map.The colors of arrows and bubble IDs signify different types of relationships in the map: black arrows represent bubbles rising, red arrows indicate bubbles splitting, and blue arrows denote bubble coalescence.If the bubble in the current frame is not connected to any earlier bubbles, it means the bubble has just been formed.For the sake of discussion, assume in Figure S2 that n = 2.All bubbles in the first frame (n -1 = 1) are sequentially numbered as, e.g., 1-8, and tagged with a bubble event label [nucleate, rise, coalesce, split] = [1, 0, 0, 0], along with a velocity matrix [V x , V y , V] = [0, 0, 0].In the second frame, all bubbles are not numbered initially.The algorithm takes the two binary images as inputs.After the algorithm has been executed, the black arrows indicate that bubbles 1, 3, 4, 6, 7, and 8 continue to rise, and so gain a bubble event tag [0, 1, 0, 0].The red arrows indicate that bubble 2 splits up into two daughter bubbles 2, and so it gains a bubble event tag [0, 0, 0, 1].For the third frame, the blue arrows indicate that bubbles 6 and 7 coalesce to form a new bubble 10; since the daughter bubbles of 2 have the tag [0, 0, 0, 1], they will keep the bubble ID 2, and an updated bubble event tag [0, 0, 1, 0].Because bubbles 5 and 8 could not be associated with a bubble in the next frame, their disappearance is associated with a death event ( × ), and because bubbles 9, 11, and 12 could not be paired with any bubble in the previous frame, their formation is associated with a birth event ( • ).The full bubble tracking algorithm is presented Table S1.
Figure S3 presents a representative result of tracking bubble dynamics across two pulsation periods, illustrating the algorithm's potential in distinguishing various bubble phenomena throughout the entire bubble lifecycle.For asymmetrical in-plane bubble splitting, smaller bubbles tend to disappear shortly after splitting, as shown in Figure S3, while the larger ones continue to ascend vertically.In the case of symmetrical in-plane bubble splitting, the binary daughter bubbles often quickly re-coalesce, typically leading to the formation of a new, large bubble.
Table S1.The bubble tracking algorithm.

Figure S1 .
Figure S1.Summary of bubble tracking results with different k 1 and k 2 for bubble coalescence

Figure S2 .
Figure S2.An illustration of the tracking algorithm.Black arrow: bubble rising.Red arrow: